Axioms for parthood and containment relations in bio-ontologies
Thomas Bittner
Institute for Formal Ontology and Medical Information Science University of Leipzig
Abstract
To fix the semantics of different kinds of parthood rela- tions we require axioms which go beyond those char- acterizing partial orderings. I formulate such axioms and show their implications for bio-ontologies. Specif- ically, I discuss parthood relations among masses, for example among body substances such as blood and portions thereof, and among components of com- plexes, for example between your stomach and your gastro-intestinal system. I contrast these with the rela- tion of being contained in (as your lungs are contained in your thorax).
The axioms considered are rooted in mereology, the formal theory of parts and wholes. By making ex- plicit the differences between the different kinds of re- lations they support different kinds of data integration in bioinformatics.
Introduction
The growth of bioinformatics has led to an increasing number of evolving ontologies which must be corre- lated with the existing terminology systems developed for clinical medicine. A critical requirement for such correlations is the alignment of the fundamental onto- logical relations used in such systems, and especially of the relation of part-of [16, 26].
However, there is one problem that stands in the way of achieving such integration: existing terminology sys- tems and ontologies are marked by an inadequate de- gree of semantic consistency at their foundations [27].
The ambiguities and inconsistencies which result from the lack of a standard unified framework for under- standing the basic ontological relationships that struc- ture these domains are an obstacle to ontology align- ment and data integration, and thus also to the sort of automatic processing of biomedical data which is the presupposition of advances in this field.
Part-whole relations play a critical role in medical con- cept representation. As Rogers and Rector [20] point out, this is most obvious in the modeling of anatomy;
but it also true of the representation of surgical pro- cedures, as well as of many physiological and disease processes, as also of the chemical pathways which lie beneath all of these.
Part-whole relations have long been the subject of ex- tensive study in philosophy [2, 24], linguistics [31], knowledge representation [10, 9], and more recently in bio-informatics [11, 22, 20, 17]. In particular, it has long been recognized that several different subtypes of the part-of relation may be identified [19, 31, 9, 13].
This recognition underlies the modeling of the part- of relation in GALEN [20] and in the Foundational Model of Anatomy (FMA) [21, 16]. All such relations are, when taken singly, treated formally as partial or- derings. However there does not exist a formal treat- ment of what distinguishes such relations one from an- other.
In this paper I give axiomatic theories for three sorts of partial ordering relations: (i) the component-of re- lation between components and the complexes they form (my mouth, my oropharynx, and my gastro- intestinal system are components of my alimentary system); (ii) part-of relations among masses such as body-substances in the sense of FMA (the blood in your left ventricle is part of the blood in your body);
and (iii) containment relations (my brain is contained in my skull, my lungs are contained in my thorax).
The formal characterization will be purely mereolog- ical and will exploit the classification of formal theo- ries given for example by Simons [23] or Varzi [29].
Thus no resources from topology or geometry are re- quired. Moreover, in all that follows I consider entities at a single moment in time. The full formal charac- terization of all the part-whole relations contained in a system like the FMA or GALEN will need to go fur- ther than what is presented here. Distinctions of the type here discussed will however be indispensable to further progress in this field.
Partial ordering structures
In this paper formal theories of different kinds of par- tial order relations are discussed. Each of the theo- ries is presented in a single-sorted first-order predi- cate logic with identity. I use the letters , and for variables. Predicates always begin with a capital letter. The logical connectors
have their usual meanings: not, identical-to, and, or, if . . . then, if and only if (iff). I write to symbol- ize universal quantification and to symbolize ex- istential quantification. Leading universal quantifiers are assumed to be understood and are omitted.
Properties of partial orderings
I introduce the binary primitive interpreted as the generic relation of proper partial ordering, i.e., stands to in the relation of proper partial ordering.
In terms of , I define the relations of (improper) par- tial order and overlap: and are in the relation of improper partial order iff either or and are identical ( ); and overlap iff they share a com- mon entity in the partial ordering hierarchy (! ):
#"%$&'( )*
+ , -!$
./
0
"
"1
I now add axioms to the effect that the relation of proper partial ordering, , is asymmetric and transi- tive (APO1-APO2).
243
,65
#+
7
243
,98
#(6:7
;<:
It then follows that proper partial ordering is irreflex- ive (TPO1) and that (improper) partial ordering " is reflexive, antisymmetric, and transitive (TPO2-4)1:
= 3
,65>
'(
= 3
,98 '"
= 3
,@?
'"( )7"AB&
= 3
,DC
'"( )7"
EA#"
Examples of partial ordering structures I now discuss three examples of partial order relations:
the component-of relation, the containment relation, and the part-of relation as it holds between masses.
The complement-of relation. Consider the component-of relation between components and complexes of my alimentary system. Figure 1 shows the component-of structure of my alimentary system according to the FMA [21]. My mouth, my orophar- ynx, and my gastrointestinal system are components
1The formal proofs are omitted here but can be obtained from the author.
of my alimentary system. In general, the nodesF and
G
in the graph structure are connected by an arrow iff entityF is a component of the complex
G
.
my Alimentary System
my Mouth my Oropharynx my Gastrointestinal system
my Upper gastrointestinal tract my Lower gastrointestinal tract
my Stomach my Duodenum my Esophagus
Figure 1: Component-of relations between the compo- nents of my Alimentary system
To see that the component-of relation satisfies the ax- ioms of proper partial orderings (APO1-2) consider that components are distinct from the complexes they form. Since my stomach is a component of my ali- mentary system, the alimentary system is not a com- ponent of my stomach. Also the alimentary system is identical to itself but not a component of itself. More- over, the component-of relation is transitive. My stom- ach is a component of my upper gastro-intestinal tract.
My upper gastro-intestinal tract is a component of my gastro-intestinal system. And also my stomach is a component of my gastro-intestinal system.
As an example for overlap of complexes consider the alimentary system and the respiratory system accord- ing to the FMA. Both have the oropharynx as a compo- nent and hence overlap in the sense of definition . Containment is the second example of a proper par- tial ordering relation. For a non-medical example consider the relation between your backpack and the books therein, or the relation between your wallet and the coins therein, or the relation between the coins and the backpack in the case where the wallet with the coins is in the backpack.
For a medical example of containment consider the re- lation which holds between my pericardial sac and my thorax in the sense that my thorax forms a container for my pericardial sac, which in turn is contained in my thorax (Figure 2). The same relation of contain- ment holds between my heart and my pericardial sac in the sense that my pericardial sac is a container for my heart. Clearly, containment understood in this sense is asymmetric and transitive. For example. The peri- cardial sac is a container for my heart, but the latter is not a container for the former. Since my heart is con-
tained in my pericardial sac and my pericardial sac is contained in my thorax, and it also holds that my heart is contained in my thorax.
my Thorax
my Pericardial sac my Lung ...
my Heart
a volume of blood
a volume of air
Figure 2: Containment relations
Notice that the interpretation of the containment rela- tion employed here is different from those in the FMA [7] and GALEN [8]. Both interpret containment as a relation between an entity and (a part of) a space that is enclosed by a container. For example for GALEN [8] the heart is contained in the mediastinum, which is a part of the thoracic space.
Here, in contrast, the relation of containment always holds between entities – the contained entity (e.g., a volume of blood) and the container (e.g., my heart).
Containers can themselves be contained in other con- tainers (e.g., my heart is contained in my pericardial sac, which in turn is contained in my thorax).
Containers have properties, like having-a-cavity, which distinguish them from non-containers. The characterization of those properties, however, is be- yond the realm of mereology. This requires at least the resources of topology and a theory of location [3, 6].
The advantage of the interpretation applied here is fourfold. Firstly, we focus on what containment means and not on what a container is. The former question can be answered within a mereological framework the latter cannot. Secondly we need only a single category in order to characterize containment – entities. In the interpretation of containment applied in the FMA and in GALEN one needs (at least) two categories: con- tained entities like the heart; and regions, like the tho- racic space, which are enclosed by container-like en- tities. Thirdly, representing containment as relation of partial order between entities allows us to characterize the similarities and differences between parthood and containment in a very explicit manner.
Fourthly, representing containment as relation be- tween entities allows us to distinguish it from the re- lation of location, which holds between entities and regions [4]. Often both relations are used in combi- nation, for example, in order to say that the heart is contained in the thorax and within the thorax it is lo- cated in a region to which we refer to as the middle
mediastinum, and which is a part of the region which is enclosed by the thorax. In general for specifying the semantics of relations in complex systems like the FMA or GALEN it is important to characterize rela- tions in separation first by employing the simplest pos- sible theory. Complex relations then can be described by combining the theories characterizing the compo- nents of the complex relation.
The parthood relation among masses is the third example of a partial ordering relation. Examples of masses are body-substances like saliva, semen, cere- brospinal fluid, inhaled air, urine, feces, blood, plasma, etc. The relation I have in mind here is the relation which holds between the blood in my body and the blood in my left ventricle. Notice that we do not have a relation of containment here. Rather names of con- tainers like ‘my body’ or ‘my left ventricle’ are used here only in order to refer to certain quantities or por- tions of the blood in my body at a certain moment in time.
One can now verify that the parthood relation among masses is a proper partial ordering relation: the blood in my heart is a proper part of the blood in my body (but not vice versa), the blood in my right ventricle is a proper part of the blood in my heart, and the blood in my right ventricle is a proper part of the blood in my body.
From these examples we can see that all three relations share the property that they form partial ordering struc- tures. Yet they are quite different in nature. It will our task in the remainder of this paper to characterize these distinctions formally.
Complexes
The characteristic property of complexes is that we can represent their partonomic structure using trees as in- dicated in Figure 1.
The formal theory of the relation component-of em- ploys a binary primitive which is interpreted as ‘the entity is a component(-part) of the entity ’.
We then add the axioms for asymmetry and transitivity for (ACP1-2)
2@3
5
+
6
2@3
8
+'%
E<
together with definitions for the improper component- of relation (which includes identity) and for component-overlap (D and D )
B"
+$
)*
,
-%$
./
0
"
7
"
1
One can see that these axioms and definitions are ex- actly analogous to what was presented in the section
on properties of partial orderings. As shown above it then follows that the component-of relation, , is irreflexive and that " is a partial ordering.
Axioms for the tree structure
We now characterize the specific character of the component-of relation beyond the fact that it has the structure of a (proper) partial ordering. We do so by adding axioms which constrain the partial order in such a way that the resulting component-of hierarchy is a finite tree structure.
For this purpose we introduce two additional predi- cates, one which holds for the root of the tree structure ( rootcp) and another which holds for atomic compo- nents, i.e., entities without a component ( Atcp).
rootcp rootcp$ / %"
Atcp Atcp$ / %
The component-of relation " is now governed by further axioms in addition to ACP1-2 (the - counterparts of APO1-2). These additional axioms fall into two groups, axioms which enforce the tree struc- ture and the finiteness of this structure respectively. We start by discussing the first group:
2@3
?
rootcp
2@3
C ,
+
B"
6:%
2@3
+
/
6
:,
ACP3 demands that every component-tree has a root.
Using the antisymmetry of " we can then prove that there exists exactly one root. This rules out the structure in Figure 3(d) from being a component-of tree.
ACP4 is a version of what I shall call the no-partial- overlap principle (NPO). It rules out the possibility of partial overlap of components by demanding that if the complexes and share a common component then either is a component of , or and are identical, or is a component of . From this it follows that cycles like the one shown in Figure 3(c) cannot occur in component-of-trees.
Notice that the no-partial-overlap principle (NPO) also rules out the possibility that two different body sys- tems which overlap (like the respiratory system and the alimentary system which share the component oropharynx) can exist within in the same component- of tree. This is because the two systems belong to dis- tinct partitions of the human body (in the sense of the theory of granular partitions [1]), which is to say to different anatomical views or perspectives.
For example, the respiratory system has as components everything that is involved in the respiration process,
and the alimentary system has as components every- thing that is is involved in the process of nutrition in- take, digestion, and excretion. Clearly, there are parts of the body which have multiple functions, and there- fore are components of different bodily systems. Each system has its own component-of tree with the partic- ular system as a whole as the root. This corresponds to the view defended by Rector et al. [18] who argue that it is an important aspect of the design of ontolo- gies to represent different views by means of separate tree structures.
root
a b
c d e f
(a) Component tree
root
a
(b) Chain
root
a b
c d f
(c) Partial overlap (cycles)
a
c d e
b
f
(d) Multiple roots
Figure 3: Component trees and non-trees.
ACP5 demands that if is a component of then there exists a component of such that and do not overlap. This rules out cases where a complex has only a single proper component. In particular, it rules out graphs like the one shown in Figure 3(b) from being representations of component-of trees. ACP5 is a ver- sion of what, following Simons [23], I call the weak supplementation principle (WSP).
The second group of axioms that characterizes the component-of relation beyond the properties of being a partial ordering are axioms which enforce the finite- ness of the component tree. ACP7 ensures that every complex has at least one atom as component. This en- sures that no branch in the tree structure is infinitely long [30, 15]. Finally ACL8 is an axiom schema which enforces that every complex is either an atom or has only finitely many components. This ensures that com-
ponent trees cannot be arbitrary broad.
Atcp
Atcp
"!#%$&$&$#')('*
!'+),-+)(
),"./01
/23/2./54
!'+),-+)(
267),'
Here 98;: =<0=>
<
/ is an abbreviation for
:
+ '
>
and?@: =<0=>
<
< for
: '
> . Extensionality
Extensionality is a property of the component-of rela- tion which tells us that two complexes are identical if and only if they have the same components. For ex- ample if the complexF : has the componentsA andB and the complexFDC has the componentsA andB then
F : andFDC are the same complex. This kind of reason- ing might seem trivial from a human perspective, but it may be very useful to enable a computer to identify and to distinguish complexes by means of their com- ponents. Moreover, when specifying the semantics of the component-of relation it is important that the prop- erty of extensionality is covered by the formalism.
In this context it is important to stress once more that we here assume an atemporal framework in which we consider reality only as it exists at a single moment in time. This means that we do not take into account the fact that a complex can have different components at different times. For example, I might lose one of my fingers but still my hand before and after the accident are the same complex. How things preserve their iden- tity while undergoing changes in this way is a difficult and controversial subject. For discussions see for ex- ample [28, 12, 14].
Given the above axioms for the component-of rela- tions, we can in fact prove that it has the property of extensionality. This is because, using ACP1, ACP2, ACP4, and ACP5, we can prove that two complexes are identical if and only if they have the same compo- nents (TCP1). Moreover using ACP6 we can prove in addition that two complexes are identical iff they have the same atomic components (TCP2).
= @3
5 /
B +
0
/
= @3
8
B&%
0 Atcp
0
"
"
1
Notice that neither TCP1 nor TCP2 is derivable from the axioms for a partial ordering alone. Both are conse- quences of the partial ordering axioms in conjunction with the specific axioms which we added in order to characterize the component-of relation. Consequently, relations which are only characterized to be a partial ordering may or may not be extensional in the sense described above. Therefore omitting axioms ACP3-7
means leaving important properties of the relation in question unspecified.
To be sure, the principles discussed here are built im- plicitly into systems like the FMA or GALEN. The important point, however, is that in order to explicate relations like component-of it is critical to make such axioms explicit.
Theorem TCP2 is also interesting from a computa- tional perspective. Clearly, when comparing com- plexes it is much easier to check the identity only of atomic components rather than of all components.
Parthood among masses
An important aspect of entities classified as masses is that they do not have any compositional structure. This means that parts can be carved out from the original mass in an arbitrary fashion. Consider, for example, body-substances like blood, plasma, urine, etc. They can be separated arbitrarily into quantities, for exam- ple, by pouring them into containers or – abstractly – by applying fiat boundaries [25]. According to the FMA [7] we can distinguish, for example, the blood in containers like my right ventricle, my artery, my coro- nary artery, and so on; we can apply fiat boundaries and distinguish the blood in the left part and the right part of my body or the blood in the upper and lower parts of my body. All these operations carve out parts or quantities of the original mass. (See also [9].) We start the formal treatment of the parthood rela- tion among masses by introducing the binary primi- tiveBFE which is interpreted as ‘the mass is a proper part of the mass ’. We then add the axioms for asymmetry and transitivity (referred to as AM1-2) together with definitions for the improper parthood re- lation and for overlap (referred to as HG and 1G , respectively) along the lines discussed in the opening paragraphs of the section on complexes. In this section we omit the statement of those axioms and definitions.
As discussed above we then can prove that " E is a partial ordering relation.
In contrast to the component-of relation, the parthood relation among masses does not form a tree struc- ture. This is because partial overlap can occur between masses. Consider, for example, the relation of over- lap between the blood in the left part of my body and the blood in the upper part of my body. They partly overlap since they share a common quantity of blood, namely the blood in the upper left part of my body, but neither is part of the other. Consequently, we can- not have the no-proper-overlap principle (NPO) as an axiom or theorem in our theory of E .
On the other hand we clearly need the weak supple- mentation principle (WSP) to be an axiom or theorem of such a theory, since WSP ensures that there cannot
be a mass that has a single proper part. Adding WSP as an axiom to this theory, however, is insufficient if we want to be able to identify and to distinguish masses in terms of their proper parts by means of a principle of extensionality similar to the one for complexes dis- cussed above. (For details on why this is the case see [23].)
In order to characterize "E beyond its structure as a partial ordering we add an axiom to the effect that if is not a part of then there exists a such that is part of and does not overlap (AM3).
2
?
B"FE %
/
"FE
', E
1
To see that AM3 is a sensible axiom consider the blood in my heart and the blood in my left ventricle. Clearly, the former is not a part of the latter. Moreover, the blood in my heart has parts, for example the blood in my right ventricle, which do not overlap with the blood in the left ventricle.
Using AM3 we can then prove the E -counterpart of the weak supplementation principle (WSP) as a the- orem (TM1), which then ensures that there cannot be a mass that has a single proper part. Using AM4 we can also prove that two masses are identical if and only if they have the same proper parts (TM2).
= 5
B E +
./
0
E 6
:, E
= 8 ./
0
E
&+
0
0
E B
E /
Consequently, the property of extensionality holds for
E
.
The theory governing the compositional structure of masses, formed by AM1-3 together with the defini- tions for " E and , E , is known in the literature as extensional mereology [23].
Containment
Consider Figure 2. Here we have a sequence of nested containers: my heart, containing a certain quantity of blood; my pericardial sac containing my heart; my tho- rax containing, among other anatomical entities, my pericardial sac. As pointed out above, containment un- derstood in this sense is irreflexive, asymmetric, and transitive.
In our theory of containment we now introduce a bi- nary primitiveB , which is interpreted as ‘the en- tity is contained in the entity ’ together, with the axioms of asymmetry and transitivity (referred to as ACT1-2). We also add the usual definitions for over- lap , and for improper containment which includes identity " , exactly analogous to those in the open- ing paragraphs of the section on complexes. We then can prove that " is a partial ordering relation.
Notice that, in contrast to the case of masses and com- plexes, we cannot here have the weak supplementation principle (WSP) either as an axiom or as a theorem in a theory of containment. This is because there are examples of containers with only one contained entity:
my brain is contained in my skull; my sister is carrying a single baby in her uterus; my pericadial sac contains my heart as the only entity, etc. Those examples would be ruled out by a theory which contained WSP.
On the other hand, our theory of containment should permit us to identify or distinguish containers – at a given point in time – by means of the entities they con- tain. We therefore add an axiom to the effect that if (i) has at least one contained entity, and (ii) every entity contained in is also contained in , then is con- tained in (ACT3).
2 = ? ./
B
0
0
B
/E< "
The idea of modeling containment using the axioms ACT1-3 is due to Brock Decker. For details see [5].
Using the definition of " and the axioms ACT1- 3 we can now prove that two containers and are identical iff they are non-empty and they contain the same entities (TCT1):
= = 5 /
0
B
1 E<B
Consider Figure 2. Like complexes, containers form tree-like structures in the sense that (1) there is a max- imal container and (2) containers do not partially over- lap. The structure is tree-like since there can be con- tainers with only a single contained entity and hence nodes with a single child node in the corresponding tree representation (as the one shown in Figure 3(b)).
Formally we define predicates for the root, rootct, and for atoms, Atct, in terms of " exactly analogous to the definitions rootcp and Atcpin the section on com- plexes.
rootct rootct
B$
1
%"
Atct Atct
B$
1
!
The root here is understood as the maximal container and atoms are understood as entities which themselves do not contain any other entities.
We then add axioms ACT4 and ACT5 in terms of rootct
"
and , exactly analogous to ACP3 and ACP4.
2 = C .
rootct
2 = ,
+
"
)+
Here ACT4 enforces the existence of a root container.
ACT5 is an instance of the no-partial-overlap principle (NPO) and rules out the partial overlap of containers.
Finally we add axioms ACT6 and ACT7 enforc- ing the condition that the resulting tree-like contain- ment structures are finite. ACT6+7 are the - counterparts of ACP6+7.
Atct
;
Atct
=!&#&$&$%$#')('*
!'+),-+)(
),=.) H
/2 /2. 4
!'+),-+)(
267 ,'
Conclusions
The theories of the component-of, mass-part-of, and contained-in relations presented in this paper share the fact that they all are partial orderings and satisfy the principle of extensionality. (In Table 1 this is indicated by the + symbols.) That the principle of extensionality is satisfied means that at a given moment in time we can identify and distinguish masses in terms of their proper parts, complexes in terms of their components, and containers in terms of the entities they contain.
The fact that this kind of reasoning is permitted, how- ever is not implied by the underlying partial ordering structure. Other principles needed to be added in or- der to support this kind of reasoning. I showed that the same principles allow us to distinguish these relations formally.
relation part. order WSP NPO EXT
component-of + + + +
mass-part-of + + +
contained-in + + +
Table 1: Theories of partial ordering relations and their underlying principles.
The principles which allow us to distinguish the three relations are the weak supplementation princi- ple (WSP) and the no-proper-overlap (NPO) principle.
The former holds in the theories of the component-of and the mass-part-of relations but not in the theory of the contained-in relation (indicated by the symbol).
The weak supplementation principle in the theories of component-of and mass-part-of tells us that a mass or a complex cannot have a single proper part or compo- nent. The no-partial-overlap principle in the theories of component-of and contained-in tells us that there cannot be partial overlap among components of com- plexes and among containers.
Acknowledgments
My thanks go to Barry Smith and Anand Kumar for helpful comments. Support from the Wolfgang Paul Program of the Alexander von Humboldt Foundation is gratefully acknowledged.
References
[1] T. Bittner and B. Smith. A theory of granular partitions. In M. Duckham, M. F. Goodchild, and M. F. Worboys, editors, Foundations of Ge- ographic Information Science, pages 117–151.
London: Taylor & Francis, 2003.
[2] H. Burkhardt and C.A. Dufour. Part/whole i:
History. In H. Burkhardt and B. Smith, editors, Handbook of Metaphysics and Ontology, pages 663 – 673. Muenchen, Philosophia, 1991.
[3] R. Casati and A.C. Varzi. Holes and Other Super- ficialities. MIT Press, Cambridge, Mass., 1994.
[4] R. Casati and A.C. Varzi. The structure of spatial localization. Philosophical Studies, 82(2):205–
239, 1995.
[5] B. Decker. Some axioms for beer glasses and backpacks. Technical report, Department of Phi- losophy, University at Buffalo, 2003.
[6] M. Donnelly. On parts and holes: The spatial structure of the human body. In Proceedings of MedInfo 2004, 2004.
[7] FMA. The Foundational Model of Anatonomy, Dec. 2003.
[8] GALEN. OpenGALEN anatonomy, version 1.7, Build 930, Feb. 2003.
[9] P. Gerstl and S. Pribbenow. Midwinters, end games, and body parts: a classification of part- whole relations. Int. J. Human-Computer Stud- ies, 43:865–889, 1995.
[10] N. Guarino, S. Pribbenow, and L. Vieu. Model- ing parts and wholes. Data & Knowledge Engi- neering, 20(3):257–258, 1996.
[11] U. Hahn, S. Schulz, and M. Romacker. Parto- nomic reasoning as taxonomic reasoning in medicine. In Proceedings of the 16th National Conference on Artificial Intelligence and 11th Innovative Applications of Artificial Intelligence Conference, pages 271–276, 1998.
[12] K. Hawley. How things persist. Oxford : Claren- don Press, 2001.
[13] I. Johansson. On the transitivity of the parthood relations. In H. Hochberg and K. Mulligan, edi- tors, On Relations and Predicates. Ontos-verlag, 2004.
[14] E. J. Lowe. A survey of Metaphysics. Oxford University Press, 2002.
[15] C. Masolo and L. Vieu. Atomicity vs. infinite di- visibility of space. In C. Freksa and D. Mark, editors, Spatial Information Theory. Cognitive and Computational Foundations of Geographic Information Science. International Conference
COSIT’99, volume 1661 of Lecture Notes in Computer Science. Springer-Verlag, 1999.
[16] J. Mejino, N. Noy, M. Musen, and C. Rosse. Rep- resentation of structural relationships in the foun- dational model of anatomy. 2003.
[17] J. L. V. Mejino, A. V. Agoncillo, K. L. Rickard, and C. Rosse. Representing complexity in part-whole relationships within the foundational model of anatomy. In Proceedings of the Amer- ican Medical Informatics Association Fall Sym- posium, 2003.
[18] A. Rector, J. Rogers, A. Roberts, and C. Wroe.
Scale and context: Issues in ontologies to link health- and bio-informatics. In Proceedings of the AMIA 2002 Anual Symposium, pages 642–
646, 2002.
[19] N. Rescher. Axioms for the part relation. Philo- sophical Studies, 6:8–11, 1955.
[20] J. Rogers and A. Rector. Galen’s model of parts and wholes: experience and comparisons. In Pro- ceedings of the AMIA Symp 2000, pages 714–8, 2000.
[21] C. Rosse and J. L. V. Mejino. A reference ontol- ogy for bioinformatics: The foundational model of anatomy. Journal of Biomedical Informatics, in press, 2003.
[22] S. Schulz and U. Hahn. Mereotopological reasoning about parts and (w)holes in bio- ontologies. In C. Welty and B. Smith, editors, Formal Ontology in Information Systems. Col- lected Papers from the 2nd International Confer- ence, pages 210 – 221, 2001.
[23] P. Simons. Parts, A Study in Ontology. Clarendon Press, Oxford, 1987.
[24] P. Simons. Part/whole ii: Mereology since 1900.
In H. Burkhardt and B. Smith, editors, Handbook of Metaphysics and Ontology, pages 673 – 675.
Muenchen, Philosophia, 1991.
[25] B. Smith. Fiat objects. Topoi, 20(2):131–48, 2001.
[26] B. Smith and C. Rosse. The role of foundational relations in the alignment of biomedical ontolo- gies. 2003.
[27] B. Smith, J. Williams, and S. Schulze-Kremer.
The ontology of the gene ontology. In Proc. An- nual Symposium of the American Medical Infor- matics Association, pages 609– 613, 2003.
[28] J. J. Thomson. Parthood and identity across time.
Journal of Philosophy, 80:201–220, 1983.
[29] A. Varzi. Parts, wholes, and part-whole rela- tions: The prospects of mereotopology. Data and Knowledge Engineering, 20(3):259–86, 1996.
[30] A. Varzi. Mereology. In Edward N. Zalta, editor, Stanford Encyclopedia of Philosophy. Stanford:
CSLI (internet publication), 2003.
[31] M.E. Winston, R. Chaffin, and D. Herrmann. A taxonomy of Part-Whole relations. Cognitive Science, 11:417–444, 1987.
